An Operator Learning Approach via Function-valued Reproducing Kernel Hilbert Space for Differential Equations

A short note on compact embeddings of reproducing kernel Hilbert spaces in $L^2$ for infinite-variate function approximation

This note consists of two largely independent parts. In the first part we give conditions on the kernel $k: Ω\times Ω\rightarrow \mathbb{R}$ of a reproducing kernel Hilbert space $H$ continuously embedded via the identity mapping into $L^2(Ω, μ),$ which are equivalent to the fact that $H$ is even compactly embedded into $L^2(Ω, μ).$ In the second part we consider a scenario from infinite-variate $L^2$-approximation. Suppose that the embedding of a reproducing kernel Hilbert space of univariate functions with reproducing kernel $1+k$ into $L^2(Ω, μ)$ is compact. We provide a simple criterion fo…

通过凸优化生成点集用于再现核希尔伯特空间中的插值

Generation of point sets by convex optimization for interpolation in reproducing kernel Hilbert spaces

本文提出通过凸优化生成点集用于再现核希尔伯特空间中的函数插值，通过构造二次锥规划问题实现生成n个点，并提出改进的序列算法逐步添加新点，数值实验表明其在多个情况下与P-贪心算法竞争。

构造性离散化与在再生核希尔伯特空间中的近似

Constructive discretization and approximation in reproducing kernel Hilbert spaces

该研究改进了再生核希尔伯特空间中离散化和近似的构造性方法，提供了更有效的样本最小二乘近似误差分析。

用解耦可分离的矩阵值核进行插值

Interpolation with uncoupled separable matrix-valued kernels

本文研究了如何利用矩阵值再生核对域Ω上的向量值函数进行近似，提出了一种解耦可分离的矩阵值核插值方法，并推导了插值误差的误差界，同时通过人工数据验证了该方法在处理向量值函数时的优势。

内部源计算逆散射：一种再生核希尔伯特空间方法

Computational inverse scattering with internal sources: a reproducing kernel Hilbert space approach

本文提出一种基于内部源逆散射问题和再生核希尔伯特空间理论，结合正则化重建非均匀介质散射势的方法，数值实验验证了其有效性。

关于再生核希尔伯特空间中不规则采样的收敛性分析

On the Convergence of Irregular Sampling in Reproducing Kernel Hilbert Spaces

本文分析再生核希尔伯特空间中不规则采样算法的收敛性，在极简假设下推导核回归的逼近性质、RKHS范数误差估计及紧域一致收敛新结果，并给出Lipschitz和Hölder连续核的收敛速率。

Reproducing kernel Hilbert spaces, polynomials and the classical moment problems

We show that polynomials do not belong to the reproducing kernel Hilbert space of infinitely differentiable translation-invariant kernels whose spectral measures have moments corresponding to a determinate moment problem. Our proof is based on relating this question to the problem of best linear estimation in continuous time one-parameter regression models with a stationary error process defined by the kernel. In particular, we show that the existence of a sequence of estimators with variances converging to $0$ implies that the regression function cannot be an element of the reproducing kernel…

Reproducing kernel Hilbert spaces in the mean field limit

Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of kernels and function spaces generated by kernels, so called reproducing kernel Hilbert spaces. Motivated by recent developments of learning approaches in the context of interacting particle systems, we investigate kernel methods acting on data with many measurement variables. We show the rigorous mean field limit of kernels and provide a detailed analysis of…

核流形：使用再生核希尔伯特空间的非线性增强降维

Kernel manifolds: nonlinear-augmentation dimensionality reduction using reproducing kernel Hilbert spaces

本文提出基于再生核希尔伯特空间学习非线性校正项的核流形降维方法，统一特征映射与径向基函数校正，具有低训练成本和误差随潜维单调下降的优势。

高斯过程与再生核希尔伯特空间：联系与等价性

Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences

本专著研究机器学习等领域中基于正定核的高斯过程概率方法与再生核希尔伯特空间非概率方法的联系与等价性，建立统一视角以衔接相关研究。